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X(2), X(4), X(67), X(69), X(316), X(524), X(671), X(858), X(2373), X(11061), X(13574), X(14360), X(14364), X(34163), X(34164), X(34165), X(34166), X(39157)

X(55838) → X(55855), X(56471) → X(56494)

Ga, Gb, Gc vertices of the antimedial triangle

Other points below

The Droussent cubic is basically the only isotomic circular pK. Its pivot is X(316) reflection of X(99) in the de Longchamps axis and isotomic conjugate of X(67).

Its singular focus F = X(10748) is the midpoint of X(4)X(14360) and the intersection of the lines X(3)X(126), X(5)X(111), X(30)X(1296).

K008 meets :

– its real asymptote (the line X(111)X(524)) at X = X(34166) on the line X(4)X(14360).

– the circumcircle at A, B, C, X(2373) and the circular points at infinity.

– the Steiner ellipse at A, B, C, X(671) and two imaginary points on the de Longchamps axis (which contains X(858) as well).

K008 is well-documented in Droussent's original paper. See the bibliography. See also Droussent central cubic, Droussent medial cubic and K007, property 8.

The isogonal transform of the Droussent cubic is K108 = pK(X32, X23) and its antigonal transform is K273.

K008 is a pivotal cubic under several conjugations :

– obviously isotomic conjugation with pivot X(316),

– X(316)-Ceva conjugation with pivot X(67),

– X(67)-cross conjugation with pivot X(55839),

– X(897)-anticomplementary conjugation with pivot X(2).

Locus properties :

1. Locus of point M whose cevian triangle is orthologic to the pedal triangle of X(23). More generally, any pK(X2,P) can be seen similarly with the pedal triangle of the isogonal of the isotomic conjugate Q of P. With Q = X(1), X(3), X(4), X(40), X(84) we obtain K034, the Lucas cubic K007, K045, K154, K133 respectively.
2. Locus of pivots of circular pKs which pass through G and X(524). See CL035.
3. Locus of P such that the trilinear polar of P with respect to the antimedial triangle is parallel to the polar of P in the conic with center X(67) which passes through Ga, Gb, Gc (Droussent).
4. Locus of P such that the circles PAGa, PBGb, PCGc are concurrent (Floor van Lamoen). More generally, for any Q which is not an in/excenter, let A'B'C' be the anticevian triangle of Q.The three circles PAA', PBB', PCC' are concurrent if and only if P lies on the circular pK with pole Q^2 (or fixed point Q) and pivot according to Special Isocubics §4.2.1. For example, with Q = K, A'B'C' is the tangential triangle and the cubic is K108.
5. Locus of P such that P, X(69) and the DF-pole of P are collinear. See CL039.
6. See also CL051.
7. Locus of pivots of circular pKs whose orthic line passes through X(524). The locus of the poles is K043. See CL035.
8. See also Walsmith triangle at K1091.

K008 has always three real prehessians P₁, P₂, P₃.

The centers of the polar conics of X(524) with respect to these prehessians are M1, M2, M3 where M1 = A X(671) /\ Ga X(69), M2 and M3 likewise.

Furthermore the polar conic of X(524) in K008 is a rectangular hyperbola.

It follows that K008 is the isogonal pK with pivot X(524) with respect to the triangle M1M2M3. Thus it must pass through the in/excenters of this latter triangle with tangents parallel to the real asymptote.

X is then the isogonal conjugate of X(524) in M1M2M3 and the singular focus F is its antipode on the circumcircle of M1M2M3.

This can easily be generalized for any point M on K008 as far as M is not a flex. If M1, M2, M3 are the centers of the polar conics of M with respect to the prehessians then K008 is a pivotal cubic with pivot M in the triangle M1M2M3. The tangents at M1, M2, M3 (and M) concur at the isopivot M' and the polar conic of M' contains these five points.

Examples :

• when M = X2, M1M2M3 is the antimedial triangle and M' is X(316).

• when M = X(67), M1M2M3 is the cevian triangle of X(316) and M' is the tangential X(55839) of X(67).

Points on K008

tP, X316/P, X67©P, X897-acP, tanP are respectively the isotomic conjugate, X(316)-Ceva conjugate, X(67)-cross conjugate, X(897)-anticomplementary conjugate, tangential of P.